Is it beneficial to represent a neural net as a matrix?
It depends on the type of neural networks you are dealing with.

For medium sized neural nets, the matrix approach is a very good way to do quick computations and even backpropogation of errors. One can even exploit sparse matrixes for understanding the sparse architecture of some neural nets.

But, for very large neural nets, using matrix computations would be computationally very intensive. So, relevant methods like graph-based stores, etc are used for them depending on the purpose and the architecture.